Continuously steerable second-order differential microphone arrays

A differential microphone array (DMA) is an important option for speech-communication applications because it can effectively reduce the influence of interference and background noise and also has the advantage of compact size (Song and Liu, 2008; Messner, 2013; Elko, 2004; Benesty and Chen, 2013). For a practical use of DMA, however, it is required to electronically steer the main beam to any desired direction. To this end, Elko (2004) introduced a steerable first-order DMA working in a three-dimensional space. After Elko's work, many related works followed (Derkx and Janse, 2009; Wu et al., 2014; Wu and Chen, 2016).

There also have been many efforts to make the second-order DMA steerable. Benesty and Chen (2013) introduced discrete steering by using six microphones in a pyramid-like arrangement. Bernardini et al. (2017) proposed a method of continuous steering of high-order DMAs by combining two reference beams. However, these methods suffer from limitations when the goal is to achieve continuous steering over the entire range of azimuthal angles. Recently, Wu and Chen (2017) proposed a steerable second-order DMA using seven microphones. Through theoretical analyses, it was shown that continuous steering can be obtained without changing the beam shape. However, they did not show how the seven-microphone configuration can be logically obtained in steering the second-order DMA.

This study presents a steerable second-order DMA. Starting with the general response of the second-order DMA, it is shown that the main lobe can be steered to any direction using dipoles allocated along the x (=0), y (=π/2), and ±π/4 axial directions. Nine- and seven-microphone-based implementation systems are proposed, in which the steered response is synthesized using back-to-back cardioids in axial directions.

When four microphones are orthogonally allocated along the x and y axes on a two-dimensional plane with an axial distance d, as shown in Fig. 1, the normalized array response of the first-order DMA steered to the direction θ_s is approximated as (Derkx and Janse, 2009)

where δ_m(θ) and D₁(θ, θ_s) denote the normalized response of the monopole and first-order dipole directed toward θ_s, respectively. In Eq. (1), d is assumed to be much smaller than the input wavelength, i.e., d ≪ λ, and the coefficients a_1,1 and a_1,2 are determined to satisfy a_1,1 + a_1,2 = 1. The dipole response in Eq. (1) is given as (Elko, 2004; Derkx and Janse, 2009)

where δ_1,x(θ) and δ_1,y(θ) are the orthogonal dipole responses with the main lobes directed toward the x and y axial directions, respectively. With the configuration shown in Fig. 1, the orthogonal dipoles have the responses

where ω is the angular frequency of the wave, τ₀ = d/c, and c is the speed of sound.

Similar to Eq. (1), the normalized response of the second-order DMA steered to the direction θ_s can be approximated as (Elko, 2004)

where $∑n=13a2,n=1$⁠, and D₂(θ, θ_s) denotes the second-order dipole directed toward θ_s. Because $D2(θ,θs)=D12(θ,θs)$⁠, we can further expand D₂(θ, θ_s) using Eq. (2) as

where $δ2,x(θ)=δ1,x2(θ)$ and $δ2,y(θ)=δ1,y2(θ)$ are the responses of the orthogonal second-order dipoles that can be easily constructed using three microphones. However, the multiplicative term δ_1,x(θ)δ_1,y(θ) cannot be directly obtained from the microphone signals.

To solve this problem, we can use the trigonometric relationship $2 sin(α cos θ)sin(α sin θ)=−cos (α(cos θ+sin θ))+cos (α(cos θ−sin θ))$⁠. Furthermore, because $cos (α(cos θ±sin θ))=1−2 sin2((α/2)(cos θ±sin θ))$ and $cos θ±sin θ=2 cos(θ∓π/4)$⁠, the multiplicative term δ_1,x(θ)δ_1,y(θ) in Eq. (5) can be rewritten as

where

The significance of Eqs. (6) and (7) is that the multiplicative response, δ_1,x(θ) δ_1,y(θ), can be obtained using second-order dipoles allocated along the θ = ±π/4 directions. It should also be noted that the microphone spacing of the dipoles in Eq. (7) is $2d/2$⁠, which is smaller than the microphone spacing d of the dipoles in Eq. (3).

Finally, R₂(θ, θ_s) in Eq. (4) can be rewritten as

Therefore, using a monopole and first- and second-order dipoles allocated along the 0, π/2, and ±π/4 directions, the response of the steered second-order DMA can be synthesized.

To improve the feasibility of synthesizing the steered response of the second-order DMA, a microphone configuration with minimal redundancy is required. A straightforward realization of Eq. (8) is shown in Fig. 2(a), where nine microphones are allocated inside a circle of radius d. Using the three microphones along the x axis, δ_1,x(θ) and δ_2,x(θ) are obtained. δ_1,y(θ), δ_2,y(θ), and δ_2,±q(θ) are similarly obtained using the microphones along the y and diagonal axes.

Using the assumption d ≪ λ, the dipole responses can be further approximated as $δ1,x(θ)≈ cos θ, δ1,y(θ)≈ sin θ, δ2,x(θ)≈ cos2θ$⁠, and $δ2,y(θ)≈ sin2θ$⁠. Likewise, Eq. (7) can be simplified as

Therefore, the dependency on both the angular frequency ω and the microphone distance d can be removed. It is then possible to obtain the configuration in Fig. 2(b), where eight microphones are uniformly distributed along a circle of radius d with one microphone at the origin. In addition, by substituting Eq. (9) in Eq. (6), we have

Therefore, two microphones can be saved because the second-order dipole along the −π/4 direction is not required anymore. As a result, we obtain a configuration based on seven microphones instead of nine microphones, as shown in Fig. 2(c). Recently, Wu and Chen (2017) proposed a seven-microphone configuration for the steerable second-order DMA. Their configuration is the same as the one in Fig. 2(c). However, they did not adequately explain how the seven-microphone configuration could be obtained.

To overcome the phase-mismatching problem that occurs when combining responses of different orders, all the responses must be made to have the same phase.

When second-order cardioids are constructed along the x, y, and ±π/4 axial directions, respectively, using the configurations in Fig. 2, their responses can be expressed as (Benesty and Chen, 2013)

where the superscript ± indicates the direction (forward or backward) the cardioid is facing. Now, the first- and second-order dipoles can be synthesized using the forward- and backward-facing cardioids, as given by

The monopole can be constructed by summing the back-to-back second-order cardioids constructed along the x and y axis as $δm(θ)=Cx+(θ)+Cx−(θ)+Cy+(θ)+Cy−(θ)$⁠.

This approach is similar to the one developed by Elko and Pong (1997) and Elko (2000), but unlike the method proposed by Wu and Chen (2017), the proposed method in Eqs. (11)–(16) does not require a phase shifter.

Using the microphone configurations in Fig. 2, computer simulations were performed. First, the white noise gains (WNGs) of each configuration were measured at a fixed frequency of 1 kHz and a microphone spacing d = 2 cm, and the results are shown in Fig. 3(a). The “Ideal” case is the result obtained from the conventional second-order cardioid, which is not steerable (Benesty and Chen, 2013). The results in Fig. 3(a) show that the direct realization of Eq. (8) [Fig. 2(a)] yields a relatively low WNG compared to the other configurations and shows a high variation with the steering angle θ_s. The smaller microphone spacing along ±π/4 directions degraded WNG performance at ±π/4 and ±3π/4 angles. On the other hand, the nine-microphone configuration produces high WGNs and shows the least variation with the steering angle.

The seven-microphone configuration has a higher WNG at some angles than the other configurations, but it shows a significant WNG drop at the angles θ_s = −π/4 and +3π/4, which is mainly caused by the removal of the microphones along the −π/4 direction. Although there is a loss of approximately 2 dB WNG at some angles compared to the Ideal case, the seven-microphone configuration is useful for practical implementation because it requires fewer microphones than the other configurations. The WNGs of the method proposed by Wu and Chen (2017) were also calculated, but they are not shown in the figure because the results were identical to those of the proposed seven-microphone configuration.

In order to confirm the preservability of the beam shape, the directivity indexes (DIs) of the three different configurations in Fig. 2 were evaluated along the steering angle. Figure 3(b) shows the results. The DIs of the three configurations have as small a variation as ±0.05 dB compared to the Ideal case, which verifies the almost constant beam shape over the entire range of azimuthal angles.

It is important to mention that the proposed method can also be used to steer any type of beam constructed using Eq. (4) with different weights. As a demonstration, a second-order hyper-cardioid was formed using the weights a_2,1 = −1/5, a_2,2 = 2/5, a_2,3 = 4/5, and the mainbeam was steered toward θ_s = 0°, 45°, 135°, and 225° using Eq. (8), respectively. The results are shown in Fig. 4, and it is seen that the beam shape is preserved regardless of the steering angle.

This study proposed a continuously steerable second-order DMA. Starting with the generic beam response, an algorithm to construct a steered beam using a monopole and dipoles at fixed directions was derived. Microphone configurations that can realize the proposed algorithm were also presented. Simulation results showed that the nine-microphone configuration yields consistent and high WNGs, but the seven-microphone configuration is favorable for practical implementation because of the marginal WNG loss.